<em>LS</em> (3)-equivalence conditions of control points and application to spatial Bézier curves and surfaces

Muhsin Incesu; Muhsin Incesu

doi:10.3934/math.2020084

AIMS Mathematics

2020, Volume 5, Issue 2: 1216-1246. doi: 10.3934/math.2020084

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Research article

LS (3)-equivalence conditions of control points and application to spatial Bézier curves and surfaces

Muhsin Incesu ^,

Department of Mathematics Education, Mus Alparslan University, Mus, 49100, Turkry

Received: 27 September 2019 Accepted: 05 December 2019 Published: 17 January 2020
MSC : 05C38, 05A15, 14L24, 51L10

Let $G$ be a transformation group and act on $X$. Any elements $x, y \in X$ are called the $G-$equivalent elements if there exist a transformation $g \in G$ such that $y = gx$ is satisfied. Similarly let $A = \left\{ x_{1}, x_{2}, ..., x_{n} \right\}$ and $B = \left\{ y_{1}, y_{2}, ..., y_{n} \right\}$ be any two subspaces of $X$ with $n-$elements. Then the subspaces $A$ and $B$ are called the $G-$equivalent subspaces if there exist a transformation $g \in G$ such that $y_{i} = gx_{i}$ is satisfied for every $i = 1, 2, ..., n$. The linear similarity transformations' group in 3 dimensional Euclidean space will be denoted by $LS(3)$. This paper presents the $G-$equivalence conditions of the subspaces $A$ and $B$ of 3-dimensional Euclidean space $E^{3}$ with $m-$elements where the transformation group $G = LS(3)$ is the linear similarity transformation group in $E^{3}$. Later the $G = LS(3)-$equivalence conditions of Bézier curves and surfaces are studied in terms of the rational $G = LS(3)$ invariants of their control points. Finally by using quadratic Bézier curves, a simple letter "S" is designed and two different shadow curves of this letter (composite curves) are obtained. Then it is emphasized that these shadow curves are $G = LS(3)-$ equivalent to designed letter "S".
- linear similarity,
- LS(3)-Equivalence,
- points systems,
- generator invariants,
- Bézier curves,
- Bézier surfaces,
- font design
Citation: Muhsin Incesu. LS (3)-equivalence conditions of control points and application to spatial Bézier curves and surfaces[J]. AIMS Mathematics, 2020, 5(2): 1216-1246. doi: 10.3934/math.2020084

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Abstract

Let $G$ be a transformation group and act on $X$. Any elements $x, y \in X$ are called the $G-$equivalent elements if there exist a transformation $g \in G$ such that $y = gx$ is satisfied. Similarly let $A = \left\{ x_{1}, x_{2}, ..., x_{n} \right\}$ and $B = \left\{ y_{1}, y_{2}, ..., y_{n} \right\}$ be any two subspaces of $X$ with $n-$elements. Then the subspaces $A$ and $B$ are called the $G-$equivalent subspaces if there exist a transformation $g \in G$ such that $y_{i} = gx_{i}$ is satisfied for every $i = 1, 2, ..., n$. The linear similarity transformations' group in 3 dimensional Euclidean space will be denoted by $LS(3)$. This paper presents the $G-$equivalence conditions of the subspaces $A$ and $B$ of 3-dimensional Euclidean space $E^{3}$ with $m-$elements where the transformation group $G = LS(3)$ is the linear similarity transformation group in $E^{3}$. Later the $G = LS(3)-$equivalence conditions of Bézier curves and surfaces are studied in terms of the rational $G = LS(3)$ invariants of their control points. Finally by using quadratic Bézier curves, a simple letter "S" is designed and two different shadow curves of this letter (composite curves) are obtained. Then it is emphasized that these shadow curves are $G = LS(3)-$ equivalent to designed letter "S".

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